Final answer:
Given that tan x = 2/3 and cos x > 0, it's determined that x is in the first quadrant. The trigonometric functions are found using the right triangle definition and the Pythagorean theorem: sin x = 2/√13, cos x = 3/√13, tan x = 2/3, cot x = 3/2, sec x = √13/3, and csc x = √13/2.
Step-by-step explanation:
To find all six trigonometric functions given that tan x = 2/3 and cos x > 0, we first identify the quadrant in which cosine is positive and tangent is also positive. This is the first quadrant, where all trigonometric functions are positive.
Since tan x = opposite/adjacent, we can represent the opposite side as 2 units and the adjacent side as 3 units for a right triangle. To find the hypotenuse, we use the Pythagorean theorem:
- Calculate the hypotenuse: hypotenuse = √(opposite² + adjacent²) = √(2² + 3²) = √(4 + 9) = √13.
- Now, we can find the six trigonometric functions based on the sides of the triangle:
- sin x = opposite/hypotenuse = 2/√13.
- cos x = adjacent/hypotenuse = 3/√13.
- tan x = opposite/adjacent = 2/3.
- cot x = adjacent/opposite = 3/2.
- sec x = hypotenuse/adjacent = √13/3.
- =csc x = hypotenuse/opposite = √13/2.
To obtain csc x and sec x in the simplest radical form, we rationalize the denominators:
- csc x = (√13/2)(√13/√13) = 13/2√13
- sec x = (√13/3)(√13/√13) = 13/3√13